Nuprl Lemma : Cauchy-Schwarz3-strict
∀n:ℕ. ∀x,y:ℕn ⟶ ℝ.
(∃i,j:ℕn. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ |Σ{x[i] * y[i] | 0≤i≤n - 1}| < (rsqrt(Σ{x[i] * x[i] | 0≤i≤n - 1}) * rsqrt(Σ{y[i] * y[i] | 0≤i≤n - 1})))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rsum: Σ{x[k] | n≤k≤m},
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
squash: ↓T,
true: True
Lemmas referenced :
Cauchy-Schwarz2-strict,
rsum_nonneg,
subtract_wf,
rmul_wf,
int_seg_wf,
subtract-add-cancel,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
square-nonneg,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
le_wf,
real_wf,
nat_wf,
exists_wf,
rneq_wf,
iff_wf,
rless_wf,
rsum_wf,
rabs_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
req_wf,
equal_wf,
square-rless-implies,
rmul-nonneg-case1,
rsqrt_nonneg,
rnexp_wf,
false_wf,
rless_functionality,
req_weakening,
req_transitivity,
rnexp-rmul,
rmul_functionality,
rsqrt-rnexp-2,
rnexp2-nonneg,
req_inversion,
rabs-rnexp,
rabs-of-nonneg,
rnexp2,
rnexp-rless,
zero-rleq-rabs,
less_than_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
functionEquality,
addLevel,
impliesFunctionality,
setEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
(\mexists{}i,j:\mBbbN{}n. x[j] * y[i] \mneq{} x[i] * y[j]
\mLeftarrow{}{}\mRightarrow{} |\mSigma{}\{x[i] * y[i] | 0\mleq{}i\mleq{}n - 1\}| < (rsqrt(\mSigma{}\{x[i] * x[i] | 0\mleq{}i\mleq{}n - 1\})
* rsqrt(\mSigma{}\{y[i] * y[i] | 0\mleq{}i\mleq{}n - 1\})))
Date html generated:
2017_10_03-AM-10_46_53
Last ObjectModification:
2017_06_19-PM-04_10_53
Theory : reals
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